Banach Limit and Fixed Point Approach in the Ulam Stability of the Quadratic Functional Equation

• 出版年: Axioms
• 主要な著者: El-sayed El-hady, Janusz Brzdęk
• フォーマット: 論文
• 言語: 英語
• 出版事項: MDPI AG 2025-03-01
• 主題: Ulam stability; quadratic functional equation; fixed point; Banach limit; normed spaces; inner product spaces
• オンライン･アクセス: https://www.mdpi.com/2075-1680/14/3/206

We show how to obtain new results on the Ulam stability of the quadratic equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo><mo>+</mo><mi>q</mi><mo>(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo>)</mo><mo>=</mo><mn>2</mn><mi>q</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>+</mo><mn>2</mn><mi>q</mi><mo>(</mo><mi>b</mi><mo>)</mo></mrow></semantics></math></inline-formula> using the Banach limit and the fixed point theorem obtained quite recently for some function spaces. The equation is modeled on the parallelogram identity used by Jordan and von Neumann to characterize the inner product spaces. Our main results state that the maps, from the Abelian groups into the set of reals, that satisfy the equation approximately (in a certain sense) are close to its solutions. In this way, we generalize several previous similar outcomes, by giving much finer estimations of the distances between such solutions to the equation. We also present a simplified survey of the earlier related outcomes.